#JavaScript [(Node.js)](https://nodejs.org), <s>125</s> 111 bytes#

-14 bytes from Zacharý

    (x)=>{z=[[],[]];l=NaN;for(i=0;i<x.length;i++){if(l>x[i])z[1].push(x[i]);else z[0].push(x[i]);l=x[i]}return z}

Basically, asks is the current item greater than the last item, add to the first list. Otherwise, add to the second.

Found out during this that comparing any number to `NaN` will always result `false`. Interesting!

Explanation:

    (x) => {                           // Create function
      z = [[], []];                    // Initialize dropsort output
      l = NaN;                         // Initialize last element
      for (i = 0; i < x.length; i++) { // For each item in input...
        if (l > x[i])                  // If current item is greater than previous
          z[1].push(x[i]);             // Then add it to the first part of output
        else                           // Elsewise
          z[0].push(x[i]);             // Add it to the nonordered part of the dropsort
        l=x[i];                        // Set last item to current item
      }                                // Repeat
      return z                         // Return finished dropsort
    }                                  // End function

[Try it online!](https://tio.run/##dc9LDsIgEAbgvaeYJUTaQH0H8QhegLBoFBWDpenDNDWevY5GjZq6mExgvvAzx/SclpvC5VWUha3tupMiDVWrS6u0NkwbI71ap2u5CwVxiku3bGJvs311kG44pBe3I37VaGdoq4WJ87o8kMdRWl9aaDX/uvTq3q@Freoig/babUJWBm9jH/bkRLRgkDCYMBgzGDGYGUrl4NdwBhFCkfRNoxlO51j4yAv2uQUDVIinmPcvRuAfBJpE/BXPWtxbH3rv8kzrM/xja/7aursB)